# Full text of "Men Of Mathematics"

## See other formats

```MEN  OF MATHEMATICS
Kronecker actually did this, at least in outline, and indicated
how the whole of algebra and the theory of numbers, including
algebraic numbers, can be reconstructed in accordance with his
demand. To get rid of V — 1, for example, we need only put a
letter for it temporarily, say i, and consider polynomials con-
taining i and other letters, say x*y,z, ... Then we manipulate
these polynomials as in elementary algebra, treating i like any
of the other letters, till the last step, when every polynomial
containing i is divided by z2 -j- 1 an<^ everything but the re-
mainder obtained from this division is discarded. Anyone ^ho
remembers a little elementary algebra may readily convince
himself that this leads to all the familiar properties of the
mysteriously misnamed ^imaginary' numbers of the text-books.
In a similar manner negatives and fractions and all algebraic
numbers (other than the positive rational integers) are elimi-
nated from mathematics - if desired - and only the blessed
V — 1 goes back to Cauchy in 1847. This was the germ of
Kronecker" s programme.
Those who dislike Kronecker's 'revolution' call it a Putsch,
which is more like a drunken brawl than an orderly revolution.
Nevertheless it has ltd in recent years to two constructively
critical movements hi the whole of mathematics: the demand
that a construction in a finite number of steps be given or
proved to be possible for any 'number' or other mathematical
'entity' whose 'existence' is indicated, and the banishment
from mathematics of all definitions that cannot be stated expli-
citly in a finite number of words, insistence upon these demands
has already done much to clarify our conception of the nature
of mathematics, but a vast amount remains to be done. As this
work is still in progress we shall defer further consideration of
it until we come to Cantor, when it will be possible to exhibit
examples.
Kronecker* s disagreement with Weierstrass should not leave
an unpleasant impression, as it may do if we ignore the rest of
Kronecker's generous life. Kronecker had no intention of
wounding his kindly old senior; he merely let his tongue run
away with him in the heat of a purely mathematical argument,
532```